B.F.
We have bought a house that was appraised for less. We agreed on a price of 168,000 and the appraisal came in at 165 000. Luckily, the sellers were nice enough to reduce the price to meet the appraised value, otherwise the 3K would have been added to the necessary downpayment, which at that point in our life would have been a problem. However, we liked the house so much that we would have paid it anyway. But unless there is something very special about your house, I don't think you can expect to get more than the appraised value. We were willing to pay more because that house backed onto the park, which in my opinion was worth the 3K.
When we sold that same house we had a relocation company help us because my husband had to move for his job. Before they put the house on the market they got 2 appraisals. One came in at 135 000 (only about 8 months after we paid 165K for it) and the other came in at 162 000. Because they were so far apart we were allowed to get a third appraisal. The third one came in at 175K. What I've learned from all that is that there can be a huge difference between different appraisers. Also, pointing out all the upgrades you did to the appraiser and having the house in good shape and clean probably helps. The third appraisal that came in at 175K I really tried to sell it to the appraiser so that it would come in as high as possible and apparently it had worked quite well.
I would talk to the realtor or the closing company about the difference, but as you said, you'll probably have to bring a check to the closing.